A solution to Comfort's question on the countable compactness of powers of a topological group

Abstract

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number α ≤ 2c , a topological group G such that G is countably compact for all cardinals γ < α, but G is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under MAcountable. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from MAcountable. However, the question has remained open for infinite cardinals. We show that the existence of 2c selective ultrafilters + 2c = 2 c implies a positive answer to Comfort’s question for every cardinal κ ≤ 2c . Thus, it is consistent that κ can be a singular cardinal of countable cofinality. In addition, the groups obtained have no non-trivial convergent sequences.